Corresponding Parts of Congruent Triangles (CPCTC)

In geometry, proving that two line segments or two angles are equal is a common task. One of the most reliable ways to do this is by using the principle of CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

What is CPCTC?

CPCTC is a fundamental rule stating that if two triangles are proven to be exactly the same size and shape (congruent), then all of their matching (corresponding) sides and angles must also be equal.

Usually, you don't start a proof with CPCTC. Instead, it serves as the crucial final step. First, you prove that two triangles are congruent using standard postulates like SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), or AAS (Angle-Angle-Side). Once the triangles are locked in as congruent, you can use CPCTC to declare that any of their corresponding parts are equal.

How to Use CPCTC in a Proof

When asked to prove that two specific segments or angles are equal, follow this three-step strategy:

1. Identify the Triangles: Look at the diagram and find two triangles that each contain one of the parts you are trying to prove equal.
2. Prove Triangle Congruence: Use the given information and geometry theorems to prove those two triangles are congruent.
3. Apply CPCTC: Conclude your proof by stating that because the triangles are congruent, the specific corresponding parts are equal.

Example Problem

Problem: In quadrilateral $ABCD$, you are given that $AB \parallel CD$ and $AB = CD$. Prove that $AD = BC$.

Proof:

1. Draw a diagonal line connecting $A$ and $C$. This creates two triangles: $\triangle ABC$ and $\triangle CDA$.
2. We know $AB = CD$ (Given).
3. Because $AB \parallel CD$, the alternate interior angles are equal: $\angle BAC = \angle DCA$.
4. Both triangles share the side $AC$, so $AC = CA$ (Reflexive property).
5. Now, we have two sides and the included angle equal in both triangles. Therefore, $\triangle ABC \cong \triangle CDA$ by the SAS (Side-Angle-Side) congruence postulate.
6. Finally, because the triangles are congruent, all their corresponding parts must be equal. Therefore, $AD = BC$ by CPCTC.